Fragment based drug design has emerged as an effective alternative to high throughput screening for the identification of lead compounds in drug discovery in the past fifteen years. Fragment based screening and optimization methods have achieved credible success in many drug discovery projects with one approved drug and many more compounds in clinical trials. The fragment based drug design starts with the identification of fragments or low molecular weight compounds that generally bind with weak affinity to the target of interest. The fragments that form high quality interactions are then optimized to lead compounds with high affinity and selectivity. The weak affinity of fragments for their target requires the use of biophysical techniques such as nuclear magnetic resonance, X-ray crystallography or surface plasmon resonance to identify hits. These techniques are very sensitive and some of them provide detailed protein fragment interaction information that is important for fragment to lead optimization. Despite the huge advances in technology in the past years, experimental methods of fragment screening suffer several challenges such as low throughput, high cost of instruments and experiments, high protein and fragment concentration requirements. To address challenges posed by experimental screening approaches, computational methods were developed that play an important role in fragment library design, fragment screening and optimization of initial fragment hits. The computational approaches of fragment screening and optimization are most useful when they are used in combination with experimental approaches. The use of virtual fragment based screening in combination with experimental methods has fostered the application of fragment based drug design to important biological targets including protein-protein interactions and membrane proteins such as GPCRs. This review provides an overview of experimental and computational screening approaches used in fragment based drug discovery with an emphasis on recent successes achieved in discovering potent lead molecules using these approaches.